I have a problem solved by Cplex that has at least two different optimal solutions with the same value.
How can I get all the different optimal solutions with the same value?
Update: it is a linear programming problem
You could as said at
https://www.ibm.com/developerworks/community/forums/html/topic?id=d8d12e36-3150-4e1c-a956-d707d17f274c&ps=25
use the solution pool or an artificial objective in order to enumerate solutions.
Related
I have a linear problem of finding all solutions that meet all constraints.
For example my variables are = [0.323, 0.123, 1.32, 6.3...]
Is it possible to get for example top 100 solutions sorted by fitness(maximization/minimization) function?
In a continuous LP enumerating different solutions is a difficult concept. E.g. consider max x, s.t. x <= 1. Obviously x=1, x=0.99999 are solutions and so are the infinite number of solutions in between. We could enumerate "corner solutions" (or basic solutions). See here for an example. Such a scheme could be adapted to find the first 100 different corner points sorted by the objective. For models with discrete variables, many constraint programming solvers will give you the possibility to find many solutions.
If you can define a fitness function as you suggested, then you might first want to solve the LP that maximizes this function. Afterwards you can include an objective cutoff that forces your second solution to be slightly worse than the first. You can implement this by introducing a cut that is your objective function with the right hand side of optimal value - epsilon.
Of course, this will not give you all (basic) solutions, but you might discover which variables are always at the same value or how much variance there is between the different solutions.
What is the best online solution to solve a linear programming problem?
I heard about several like Gurobi.
One thing I especially want is the possibility to get an approximate solution when the exact resolution takes too long.
The most comprehensive online optimization system is NEOS. It takes models in a variety of input formats and has a wide range of solvers.
Many solvers have settings to allow them to terminate early, even before optimality is reached, if you want an approximate and quick solution. But often your best bet in that case is to use a heuristic algorithm designed specifically for your problem.
I have a set of problems (sets of equations and inequalities) for which I know that all variables have to be integers, and have finitely many solutions. I know that if I take any random objective function and let an lp or mip solver onto it, it finds a solution, however I want all solutions to the problem, and of course, as efficiently as possible. I don't really care about optimizing anything, but apparently most of the software that deals with it does. Is there any solver that can do that? If so, which one is the best/simplest one, or which one would you recommend? At best one that can be used as a C/C++ library.
There is a nice blog post by Paul Rubin on how to find K best solutions, which can be easily generalized to get all the solutions. As Ali suggested one of the approaches is to use a solution pool. Two other approaches are:
Use an incumbent callback to track and reject solutions.
Use an incumbent callback with solution injection.
See the blog post for details.
IBM ILOG CPLEX has a solution pool feature and it's free for academic purposes.
I guess you can probably get all solution if you set the maximum pool size sufficiently large. I don't know for sure, never tried.
I have an integer linear optimisation problem and I'm interested in feasible, good solutions. As far as I know, for example the Gnu Linear Programming Kit only returns the optimal solution (given it exists).
This takes endless time and is not exactly what I'm looking for: I would be happy with any good solution, not only the optimal one.
So a LP-Solver that e.g. stops after some time and returns the best solution he found so far, would do the job.
Is there any such software? It would be great if that software was open source or at least free as in beer.
Alternatively: Is there any other way that usually speeds up Integer LP problems?
Is this the right place to ask?
Many solvers provide a time limit parameter; if you set the time limit parameter, they will stop once the time limit is reached. If an integer feasible solution has been found, it will return the best feasible solution found to that point.
As you may know, integer programming is NP-hard, and there is a real art to finding optimal solutions as well as good feasible solutions quickly. To compare the different solvers, see Prof. Hans Mittelmann's Benchmarks for Optimization Software. The MILP benchmarks - particularly MIPLIB2010 and the Feasibility Benchmark should be most relevant.
In addition to selecting a good solver, there are many things that can be done to improve solve times including tuning the parameters of the solver and model reformulation. Many people in research and industry - including myself - spend our careers working on improving the solve times of MIP models, both in general and for specific models.
If you are an academic user, note that the top commercial systems like CPLEX and Gurobi are free for academic use. See the respective websites for details.
Finally, you may want to look at OR-Exchange, a sister site to Stack Overflow that focuses on the field of operations research.
(Disclaimer: I currently work for Gurobi Optimization and formerly worked for ILOG, which provided CPLEX).
If you would like to get a feasibel integer solution fast and if you don't need the optimal solution, you can try
Increase the relative or absolute Gap. Usually solvers have small gaps of say 0.0001% for relative gap. This means that the solver will continue searching for MIP solutions until it the MIP solution is not farther than 0.0001% away from the optimal solution. Increase this gab to e.g. 1%., So you get good solution, but the solver will not spent a long time in proving optimality.
Try different values for solver parameters concerning MIP heuristics.
CPLEX and GUROBI have parameters to control, MIP emphasis. This means that the solver will put more emphasis on looking for feasible solutions or on proving optimality. Set emphasis to feasible MIP solutions.
Most solvers like CPLEX, Gurobi, MOPS or GLPK have settings for gap and heuristics. MIP emphasis can be set - as far as I know - only in CPLEX and Gurobi.
A usual approach for solving ILP is branch-and-bound. This utilized the solution of many sub-LP (without-I). The finally optimal result is the best of all sub-LP. As at least one solution is found you could stop anytime and would have a best-so-far.
One package that could do it, is the free lpsolve. Look there at set_timeout for giving a time limit, and when it is ILP the solve function can return in SUPOPTIMAL the best_so_far value.
As far as I know CPLEX can. It can return the solution pool which contains primal feasible solutions in the search, and if you specify the search focus on feasibility rather on optimality, more faesible solutions can be generated. At the end you can just export the pool. You can use the pool to do a hot start so it's pretty up to you. CPlex is free now at least in my country as you can sign up as a researcher.
Could you take into account Microsoft Solver Foundation? The only restriction is technology stack that you prefer and here you should use, as you guess, Microsoft technologies: C#, vb.net, etc. Here is example how to use it with Excel: http://channel9.msdn.com/posts/Modeling-with-Solver-Foundation-30 .
Regarding to your question it is possible to have not a fully optimized solutions if you set efficiency (for example 85% or 0.85). In outcome you can see all possible solutions for such restriction.
I need to solve a few mathematical equations in my application. Here's a typical example of such an equation:
a + b * c - d / e = a
Additional rules:
b % 10 = 0
b >= 0
b <= 100
Each number must be integer
...
I would like to get the possible solution sets for a, b, c, d and e.
Are there any libraries out there, either open source or commercial, which I can use to solve such an equation? If yes, what kind of result do they provide?
Solving linear systems can generally be solved using linear programming. I'd recommend taking a look at Boost uBLAS for starters - it has a simple triangular solver. Then you might checkout libraries targeting more domain specific approaches, perhaps QSopt.
You're venturing into the world of numerical analysis, and here be dragons. Seemingly small differences in specification can make a huge difference in what is the right approach.
I hesitate to make specific suggestions without a fairly precise description of the problem domain. It sounds superficiall like you are solving constrained linear problems that are simple enough that there are a lot of ways to do it but "..." could be a problem.
A good resource for general solvers etc. would be GAMS. Much of the software there may be a bit heavy weight for what you are asking.
You want a computer algebra system.
See https://stackoverflow.com/questions/160911/symbolic-math-lib, the answers to which are mostly as relevant to c++ as to c.
I know it is not your real question, but you can simplify the given equation to:
d = b * c * e with e != 0
Pretty sure Numerical Recipes will have something
You're looking for a computer algebra system, and that's not a trivial thing.
Lot's of them are available, though, try this list at Wikipedia:
http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems
-Adam
This looks like linear programming. Does this list help?
In addition to the other posts. Your constraint sets make this reminiscent of an integer programming problem, so you might want to check that kind of thing out as well. Perhaps your problem can be (re-)stated as one.
You must know, however that the integer programming problems tends to be one of the harder computational problems so you might end up using many clock cycles to crack it.
Looking only at the "additional rules" part it does look like linear programming, in which case LINDO or a similar program implementing the simplex algorithm should be fine.
However, if the first equation is really typical it shows yours is NOT a linear algebra problem - no 2 variables multiplying or dividing each other should appear on a linear equation!
So I'd say you definitely need either a computer algebra system or solve the problem using a genetic algorithm.
Since you have restrictions similar to those found in linear programming though you're not quite there, if you just want a solution to your specific problem I'd say pick up any of the libraries mentioned at the end of Wikipedia's article on genetic algorithms and develop an app to give you the result. If you want a more generalist approach, then you've got to simulate algebraic manipulations on your computer, no other way around.
The TI-89 Calculator has a 'solver' application.
It was built to solve problems like the one in your example.
I know its not a library. But there are several TI-89 emulators out there.